Chapter 10: Q. 13 (page 668)

A random sample of 200 New York State voters included 88 Republicans, while a random sample of 300 California voters produced 141 Republicans. Which of the following represents the $95 %$ confidence interval that should be used to estimate the true difference in the proportions of Republicans in New York State and California?
(a) $(0.44 - 0.47) \pm 1.96 \frac{(0.44) (0.56) + (0.47) (0.53)}{\sqrt{200 + 300}}$
(b) $(0.44 - 0.47) \pm 1.96 \frac{(0.44) (0.56)}{\sqrt{200}} + \frac{(0.47) (0.53)}{\sqrt{300}}$
(c) $(0.44 - 0.47) \pm 1.96 \sqrt{\frac{(0.44) (0.56)}{200} + \frac{(0.47) (0.53)}{300}}$
(d) $(0.44 - 0.47) \pm 1.96 \sqrt{\frac{(0.44) (0.56) + (0.47) (0.53)}{200 + 300}}$
(e) $(0.44 - 0.47) \pm 1.96 \sqrt{(045) (0.55) (\frac{1}{200} + \frac{1}{300})}$

Short Answer

Expert verified

$(c) (0.44 - 0.47) - 1.96 \cdot \sqrt{\frac{0.44 (0.56)}{200} + \frac{0.47 (0.53)}{300}}$

Step by step solution

Given Information

$n_{1} = 200$

$x_{1} = 88$

$n_{2} = 300$

$x_{2} = 141$

Explanation

The formula confidence interval of difference of proportions:

$({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z_{α / 2} \cdot \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$

The sample proportion is calculated by dividing the number of successes by the sample size.:

${\hat{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{88}{200} = 0.44$

${\hat{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{141}{300} = 0.47$

Determine the critical value using the table $z$ :

localid="1650359734870" $z_{α / 2} = z_{0.025} = 1.96$

The confidence interval is then:

localid="1654604168020" $(0.44 - 0.47) \pm 1.96 \cdot \sqrt{\frac{0.44 (1 - 0.44)}{200} + \frac{0.47 (1 - 0.47)}{300}}$

localid="1654604178645" $= (0.44 - 0.47) \pm 1.96 \cdot \sqrt{\frac{0.44 (0.56)}{200} + \frac{0.47 (0.53)}{300}}$

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